This disclosure relates to the field of analysis of physical properties of porous materials. More specifically, the disclosure relates to the methods for measuring the property of permeability of porous materials. Permeability is an important physical property for many fields of study. As related to this disclosure, one important field of study involving porous materials is the petroleum industry. Characterization of Earth formations is important to the industry particularly with reference to the ability to predict fluid production from such formations.
Darcy's Law is the fundamental law describing fluid flow through a porous medium. Much of the petroleum industry uses what is called the steady state method to measure permeability. This is a direct application of Darcy's Law. The steady state method involves setting a pressure differential across a sample of a formation. Because of this pressure differential, fluid will flow through the porous matrix of the sample. By measuring the flow rate (Q), the pressure at both ends of the sample (i.e., the pressure drop—ΔP), and the sample geometry (L and A), it is simple to apply Darcy's Law to the measured flow rate, pressure drop and sample geometry to obtain the permeability (k). μ is the viscosity of the fluid, which is known or can be measured.
                    k        =                              -                                          L                *                Q                            A                                ⁢                      μ                          (                              Δ                ⁢                                                                  ⁢                P                            )                                                          (        1        )            
Permeability can also be measured using techniques typically described as pressure transient techniques. Pressure transient techniques typically involve measuring the response to a transient in pressure where the sample is configured such that its permeability influences the response. An example of a boundary value problem used is when a cylindrical sample of known length and diameter is jacketed in an impermeable membrane and placed between two endcaps providing pore pressure access to each end of the sample. At one end of the sample, typically called the upstream end, a pore pressure is controlled or otherwise known. At the other end of the sample, typically called the downstream end, a known boundary condition is established such that the relationship between the volumetric flow through the boundary and pressure at the boundary is known. Within the sample, a governing equation is specified that relates pressure to flow within the sample where permeability is one of the parameters defining the governing equation. A common governing equation combines Darcy's law with the mass balance equation and can be written as:(k/η)∂2p/∂x2=S∂p/∂t  (2)where k is the sample permeability, η is the fluid viscosity, p is the pressure, x is the position within the sample, S is the specific storage, and t is time.
One transient method known in the art is the complex transient method. In a common implementation of the complex transient method, the pressure at the upstream end of the sample of the sample is controlled, and the downstream end of the sample is connected to a known volume of pore fluid. The pressure at the downstream end of the sample is monitored as a function of time. The transfer function describing the relationship between the perturbation in pressure at the upstream end of the sample and the response measured at the downstream end of the sample is a function of the length (L), cross-sectional area (A), permeability (k), and specific storage (S) of the sample, the viscosity (η) and compressibility (β) of the fluid, and the volume (V) in communication with the bottom of the sample. Provided that the sample dimensions, and fluid properties are known, the permeability and specific storage of the sample can be determined by measurement of the pressure response at the downstream end of the sample in response to the controlled pressure at the upstream end of the sample.
In the absence of specific storage of the sample, or equivalently if the storage capacity of the sample is small compared to the storage capacity of the down-stream reservoir, the governing equation simplifies to that of Darcy's Law. In this limiting case, the measurement of permeability is relatively straightforward using accepted analytical solutions.
For the case where specific storage of the sample is sufficiently large to measurably influence the pressure response, the analysis becomes more difficult. While analytical solutions to the re-equilibration process are available in a number of forms, in the general case there is no closed-form expression to compute permeability from the equilibration process. In such cases, the permeability and specific storage are determined by iterative comparison with model predictions. The permeability and specific storage are determined by finding the values of permeability and specific storage that produce the best fit to the measured response.
A common assumption relating the specific storage to the fluid properties (e.g. fluid compressibility (β) and rock properties is to assume that S=(α+βϕ) where α is the pore volume compressibility of the rock and ϕ, is the porosity of the rock. If α and β are assumed known, the specific storage can be directly related to the sample porosity.
The complex transient method has several possible advantages over the steady state method and other transient methods.
First, the complex transient method is faster than the traditional steady-state flow technique. This is especially important for low permeability samples. As the permeability of the sample decreases, the required duration of a permeability test increases.
Second, the complex transient method can measure permeability without creating any net flow through the sample. This is preferable for some samples which would incur changing physical properties during large amounts of flow.
Third, the complex transient method allows for the possibility to measure the specific storage of the sample. The complex transient method makes use of a transient pressure response that in some cases is sensitive to both the permeability and the specific storage. When this is the case, the complex transient method allows for the simultaneous measurement of both permeability and specific storage.
Fourth, execution of a complex transient measurement can be optimized for a sample in response to the sample's permeability without the need to adjust the measurement apparatus. In the complex transient method, this optimization is performed by adjusting the timescale of the pressure transient controlled at the upstream end of the sample in order to obtain a desired response to the transient at the downstream end.
A requirement of the complex transient method known in the art is that the user needs to specify the time scale for the transient before performing the measurement. This is known to be performed largely on a trial and error basis: A measurement is performed with an arbitrarily chosen (or otherwise guessed) time scale; the quality of the measurement is analyzed; then the measurement is repeated using an adjusted time scale. Having an automated technique to determine an appropriate time scale as part of the measurement process itself may facilitate use of the complex transient method.